Chapter 9: Q. 9.7 (page 412)

A transition probability matrix is said to be doubly
stochastic if
$\sum_{i = 0}^{M} P_{i j} = 1$
for all states j = 0, 1, ... , M. Show that such a Markov chain is ergodic, then
j = 1/(M + 1), j = 0, 1, ... , M.

Short Answer

Expert verified

It is proved that a Markov chain is ergodic, then $π_{j} = \frac{1}{M + 1}$ for $j = 0, 1, . . ., M$

Step by step solution

Given Information

We have given that the transition probability matrix is doubly stochastic if

$\sum_{i = 0}^{M} P_{i j} = 1$

for all states $j = 0, 1, . . ., M$ .

Simplify

As the chain ergodic and the transition matrix is doubly stochastic, there exists a unique stationary distribution $π$ . Now, we just have to check that is that localid="1648139523807" $π_{i} = \frac{1}{m +!}$ solution of the system of the equation $π = πP$ i.e., is it true

$π_{j} = \sum_{i} p_{ij} π_{j}$ $π_{j} = \sum_{i} p_{i_{j}} π_{j}$

but, we havelocalid="1648139404599" $\sum_{i} p i j = 1$ , which means

$\frac{1}{M + 1} = \frac{1}{M + 1} . \sum_{i} p i j = \frac{1}{M + 1} . 1 = \frac{1}{M + 1}$

So, we have proved that the stationary distribution is localid="1651480780070" $π_{j} = \frac{1}{m + 1}$ .

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A transition probability matrix is said to be doubly
stochastic if
$\sum_{i = 0}^{M} P_{i j} = 1$
for all states j = 0, 1, ... , M. Show that such a Markov chain is ergodic, then
j = 1/(M + 1), j = 0, 1, ... , M.

Short Answer

Step by step solution

Given Information

Simplify

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A transition probability matrix is said to be doublystochastic if∑i=0MPij=1for all states j = 0, 1, ... , M. Show that such a Markov chain is ergodic, thenj = 1/(M + 1), j = 0, 1, ... , M.

Short Answer

Step by step solution

Given Information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Probability and Statistics

Pure Maths

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

A transition probability matrix is said to be doubly
stochastic if
$\sum_{i = 0}^{M} P_{i j} = 1$
for all states j = 0, 1, ... , M. Show that such a Markov chain is ergodic, then
j = 1/(M + 1), j = 0, 1, ... , M.